John is correct that iterated subtraction gives -a*b if it is performed
in the same manner as iterated addition is used to give a*b. However,
just saying "iterated addition gives multiplication" is not by itself an
explanation of the procedure.
When doing iterated addition to achieve the multiplication a*b, one of a
or b is the addend while the other is the number of iterations. Let's
say b is the addend. We start with accumulator = zero and add b into
the accumulator a times. At that point the accumulator contains the
product. Thus, we have to have a starting value for the accumulator
(zero), and addend (b), and we have to keep track of the iterations
until (a) iterations have been performed.
Likewise there is a procedure for doing division by iterative
subtraction. It is not the exact inverse of iterative addition, but it
is close.
When doing iterative subtraction to achieve a/b, the starting
accumulator is a, the subtractand is b, and we again have to keep track
of the iterations. The quotient is [the number of iterations required
for the accumulator to hit zero] or [number of iterations required to
cross zero, minus one]. [The absolute value of the final accumulator]
is the remainder.
It is not obvious to me why these two processes cannot be construed as
somewhat inverses of each other. Exact inverses... no. Somewhat
inverses or somewhat parallel ideas... I think so. The accumulator
either starts at zero and ends up as the product, or it starts at the
dividend and ends up as the negative-remainder. Number b is either the
addend or the subtractend. The number of iterations is a or is the
quotient or [quotient minus one]. It seems to me the differences (aside
from subtraction versus addition, and division versus multiplication)
are how the accumulator, variables, number of iterations are used and/or
start and end.
I did enough assembly-language programming in the 1970s to know that
iterative subtraction was the easiest way to do division.
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu